We are given a triangle $ABD$ with angle $A = 62^\circ$, angle $B = 14^\circ$, and angle $D = 31^\circ$. We need to find the reflex angle $BCD$.

1. Problem Description

We are given a triangle $ABD$ with angle $A = 62^\circ$, angle $B = 14^\circ$, and angle $D = 31^\circ$. We need to find the reflex angle $BCD$.

2. Solution Steps

First, we find the angle $ADB$.

The sum of angles in a triangle is $180^\circ$. Therefore, in triangle $ABD$,

$A + B + D = 180^\circ$

$62^\circ + 14^\circ + D = 180^\circ$

$76^\circ + D = 180^\circ$

$ADB = 180^\circ - 76^\circ = 104^\circ$.

Next, we observe that points $A, C, B$ are collinear, implying that $ACB$ is a straight line, meaning that angle $ACB = 180^\circ$. Also $C, D, A$ are collinear, implying that $CDA$ is a straight line, meaning that angle $CDA = 180^\circ$.

In triangle $BCD$, the sum of the angles is $180^\circ$, i.e., $DBC + BCD + CDB = 180^\circ$.

Therefore, $14^\circ + BCD + 31^\circ = 180^\circ$.

$45^\circ + BCD = 180^\circ$

$BCD = 180^\circ - 45^\circ = 135^\circ$.

However, the question asks for the reflex angle $BCD$, which is the angle greater than $180^\circ$.

The reflex angle $BCD = 360^\circ - \angle BCD$.

Reflex angle $BCD = 360^\circ - (180^\circ - (14^\circ + \angle BDC)) = 360^\circ - (180^\circ - (14^\circ + \angle BDC))$.

Since the angles $BCD$ and $BCA$ form a straight line, we have $\angle BCA = 180^\circ$.

Since $ACD$ is a straight line, we know that $\angle ACD = 180^\circ$.

Also, $\angle ACB$ is a straight line, so $\angle ACB = 180^\circ$.

Therefore $\angle ACB = \angle ACD = 180^\circ$.

Since we have a triangle $ABD$, we know that $\angle ADB = 180^\circ - (62^\circ + 14^\circ) = 180^\circ - 76^\circ = 104^\circ$.

The angle at $C$ can be found by looking at angles around the point $C$.

Let the angle $ACD = x$ and the angle $BCD = y$.

Also, $ACD + BCD = 360$.

Then $x + y = 360$.

So, the reflex angle $BCD = 360 - x$.

We have the triangle $ABC$. We have $A=62$, $B=14$, so $C=180-(62+14) = 180 - 76 = 104$.

$BCD = 104+62+14+31=180-76=104$

We know that $\angle BCA + \angle BCD = 360^\circ$.

First, calculate $\angle BCA$. $\angle BCA = 180 - \angle BAC - \angle ABC = 180 - 62 - 14 = 104$.

Now, we have $\angle BCD = 180 - (\angle CBD + \angle CDB) = 180 - (14 + 31) = 180 - 45 = 135$.

The reflex $\angle BCD = 360 - 135 = 225^\circ$.

3. Final Answer

225